How to merge faces without creating new shapes
If you create 2 unfilled circles and press F, they will fill independently, Blender won’t fill the gap between them. If you put the small circle inside the big one, it’s the exact same situation, the same thing will happen.


This explains what’s happening with your ring, if you delete all the faces within the ring, Blender can’t guess the 2 circles are supposed to be linked to each other, it will fill both the outer and the inner circle but won’t fill the gap between them.
To do so you need to create at least 2 edges between the outer and the inner circle , then in Edge Select mode select each face and fill.